# -*- coding: utf-8 -*-
"""
Created on Sat Dec 12 19:03:48 2020

@author: neil
"""

# calculating pi, 
# To illustrate basic python loops,
# The greeks used inscribed and circumscribed polygons to estimate pi to a couple of decimal places
# With the computer, we can approximate pi by inscribing ever increasing polygons
# (the greeks had difficulty with the sheer number of calculations, we let the computer do it)
# the key, as Archimedes realized, was that using Pythagoras's theorm, he derived a recursive
# formula for the side length of a regular polygon, based on the side length of a n/2 polygon:
# Sn = sqrt( 1/2 - sqrt( (1/2)**2 - (Sm/2)**2 ) ) 
# where Sn is the side length of an n-sided polygon, and Sm is the side length of an (n/2) polygon
# this is a messy formula, which meant the greeks couldn't get much accuracy calculating by hand
# below we use this formula (slighly rearranged to reduce subtraction errors)

# import extra 'modules', this is typical if you want to do anything fancy
# we don't need it much here, but we will import 'numpy' as our basic fancy math resource
import numpy as np

# the initial part of any program usually is where you put any variables etc, that
# need to be set, or are likely to be changed by the user
# since we are going to iterate to find pi, set a max number of iterations 
maxiter = 30

# start with an inscribed square (a 4-gon), since we know the edge length
si = np.sqrt(2)           # side of a inscribed square (si = n-gon) inside a radius 1 circle

print("n-gon: {0:10d} - {1:1.18f}".format(4,2*si) )               
for n in range(1,maxiter):  # 
    t =  (si/2)**2                                                  # si here is the old side length
    si = ( np.sqrt( ((1-np.sqrt(1-t))**2 + t) ) )                   # si is the new side length
    print("n-gon: {0:10d} - {1:1.18f}".format(4*2**n,si*2**(n+1)) ) # multi by number of sides, 2**(n+1)
                                                                    # to get total circumference
    
# An interesting point is that tiny erros acumulate, resulting in errors at the
# 15 decimal place 
pie = np.pi               # get an accurate (to at least 15 places) value for pi
print("Pythons default pi  {0:1.18f}".format(pie) )  # the numpy module has its own accurate value for pi
# major side note, python's value of pi has an error in the 16th place,
# this is due to storing np.pi in a single precision storage location

# print("Exact value pi      {0:1.18f}".format(3.141592653589793238) )  # doesn't work
print("Exact value pi      {0:1.15f}{1:3d}".format(3.141592653589793,238) ) 
# python defaults to its 'single precision' which is valid to about 14 or 15 significant
# digits. If you really need more you need to import multiprecison "mpmath" module
# otherwise variables are stored in containers that truncate